QUANTIZED HYPERALGEBRA OF RANK 1 W. Chin and L. Krop†

We study the algebra U ζ obtained via Lusztig's 'integral' form [Lu 1, 2] of the generic quantum algebra for the Lie algebra g = sl 2 modulo the two-sided ideal generated by K l − 1. We show that U ζ is a smash product of the quantum deformation of the restricted universal enveloping algebra u ζ of g and the ordinary universal enveloping algebra U of g, and we compute the primitive (= prime) ideals of U ζ. Next we describe a decomposition of u ζ into the simple U-submodules, which leads to an explicit formula for the center and the indecomposable direct summands of U ζ. We conclude with a description of the lattice of cofinite ideals of U ζ in terms of a unique set of lattice generators. §0 Introduction G. Lusztig constructed in [Lu 1,2,3] quantum algebras associated to the defining relations of the finite-dimensional semi-simple Lie algebra g. The method used there is similar to the one employed by Kostant [Ko] in his construction of the hyperalgebra for g. We refer to Lusztig's algebra as the quantum hyperalgebra of g. Let g = sl 2 be the rank 1 simple Lie algebra. Fix a field k of characteristic zero containing a primitive ℓ-th root of unity ζ of an odd order. We let U q stand for the usual generic quantum algebra of sl 2. We letˆU ζ denote the quantum algebra associated with sl 2 as in [Lu 1]. We define U ζ as the quotient ofˆU ζ modulo the ideal generated by K ℓ − 1. We denote by u ζ the Frobenius-Lusztig kernel in U ζ and let U stand for the ordinary enveloping algebra of sl 2 over k. The goal of this paper is to obtain an explicit description of the primitive ideals, the center, blocks and the lattice of cofinite ideals of U ζ .